In many practical scenarios, such as hand-held cameras or ones mounted on a moving vehicle, it is difficult to eliminate camera shake. Sensor movement during exposure leads to unwanted blur in the acquired image. Under the assumption of white noise and spatially-invariant blur across the sensor this process may be modeled by Eq. (1) below:B(x)=(I*k)(x)+η(x)  (1)where * denotes the convolution operation, B is the acquired blurry image, k is the unknown blur kernel and η(x) is a zero-mean, identically- and independently-distributed noise term at every pixel x=(x,y). Blind image deconvolution, the task of removing the blur when the camera motion is unknown, is a mathematically ill-posed problem since the observed image B does not provide enough constraints for determining both I and k. Most deblurring techniques, therefore, introduce additional constraints over I and k. The most common framework for incorporating such prior knowledge is through maximum a posteriori (MAP) estimation. Norms favoring sparse derivatives are often used to describe I as a natural image. While not being failure-free, this approach was shown to recover very complex blur kernels and achieve impressive deblurred results. However, the maximization of these estimators is a time consuming task involving the computation of the latent image multiple times.
An alternative approach to the problem, which did not receive as much attention, extracts the blur kernel k directly from the blurry image B without computing I in the process. The basic idea is to recover k from the anomalies that B shows with respect to the canonical behavior of natural images. One solution known in the art is to compute the 1D autocorrelation of the derivative of B along the sensor movement direction. Normally, image derivatives are weakly correlated and hence this function should be close to a delta function. The deviation from this function provides an estimate for the power spectrum (PS) of the kernel, |{circumflex over (k)}(ω)|2.
One known approach is to recover the PS of two-dimensional kernels and use the eight-point Laplacian for whitening the image spectrum. The blur kernel is then computed using a phase retrieval technique that estimates the phase by imposing spatial non-negativity and compactness. This approach consists of evaluating basic statistics from the input B and, unlike methods that use MAP estimation, does not involve repeated reconstructions of I. While this makes it favorable in terms of computational-cost, the true potential of this approach in terms of accuracy was not fully explored.
Current solutions were directed at the removal of image blur due to camera motion. Blind-deconvolution methods that recover the blur kernel k and the sharp image I rely on various regularities natural images exhibit. The most-dominant approach for tackling this problem, in the context of spatially-uniform blur kernel, is to formulate and solve a MAP problem. This requires the minimization of a log-likelihood term that accounts for Eq. (1) plus additional prior terms that score the resulting image I and kernel k. One solution uses an autoregressive Gaussian prior for I(x) and another solution uses a similar Gaussian prior over high-frequencies (derivatives) of I. Both priors are blind to the phase content of k and are not sufficient for recovering it. Another approach further assumes that the blur is symmetric (zero phase) while another approach incorporates adaptive spatial weighting which breaks this symmetry. Yet in another known approach, the Gaussian image prior is replaced with a Laplace distribution defined by the l1 norm over the image derivatives. This choice is more consistent with the heavy-tailed derivative distribution observed in natural images. Yet another suggestion of the current art shows that this prior is not sufficient for uniqueness and may result in degenerate delta kernels. Indeed, methods that rely on sparse norms often introduce additional constraints such as smoothness of the blur-kernel and two motion-blurred images or use alternative image priors such as spatially-varying priors and ones that marginalize over all possible sharp images I(x).